Question

Let $x_1,x_2,\cdots ,x_n$ be n observations, and let $\overline{x}$ be their arithmetic mean and ${\sigma }^2$ be the variance.
Statement – 1: Variance of $2x_1,2x_2,\cdots ,2x_n\text{is}4{\sigma }^2$
Statement – 2: Arithmetic mean $2x_1,2x_2,\cdots ,2x_n\text{is}4\overline{x}$

• A. Statement – 1 is false, Statement – 2 is true
• B. Statement – 1 is true, statement – 2 is true; statement – 2 is a correct explanation for Statement – 1.
• C. Statement – 1 is true, statement – 2 is true; statement – 2 is not a correct explanation for Statement – 1.
• D. Statement – 1 is true, statement – 2 is false.

Answer 1

The correct option is D

Statement – 1 is true, statement – 2 is false.

If each observation is multiplied by k, mean gets multiplied by k and variance gets multiplied by $k^2$. Hence the new mean should be $2\overline{x}$ and new variance should be $2^2{\sigma }^2$ .
So statement –I s true and statement – 2 is false.